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Full text of "Treatise On Analysis Vol-Ii"

278      XIV   INTEGRATION IN LOCALLY COMPACT GROUPS

PROBLEMS

1.    On the additive group R, let A be Lebesgue measure, let JJL = 9?!  A, where I is the interval
[0 + oo] and let a < b be two distinct points of R. Show that the convolution product
((sfl __ fr) * fj) * A is defined, but that p and A are not convolvable. Show that the
convolutions p, * ((* - *) * ^)and (/* * ( ~~ ^ * ^ are both defined and are unequal.

2.    Let G be a locally compact group which is not unimodular.

(a)    Show that there exists a bounded positive measure /z on G such that AG  p,
is not bounded (take /x to be discrete).

(b)    Let A be a left Haar measure on G. By (14.6.3), /x and A are convolvable. Show
that A and JJL are not convolvable.

3.    Let G be a locally compact group.

(a)    Let /z, v be two positive measures on G. If ^ * v = se, show that /x = aex and
v = a~lex-1 for some x e G and a ^ 0 (cf. 14.5.4).

(b)    Give an example of a positive measure on the group Z/2Z whose support is the
whole group and which has an inverse (with respect to convolution product).

4.    (a)   In the set M(R), consider the two sequences of bounded measures /xn = en and
vn = f-n, both of which tend to 0 in the topology ^~2 defined in Section 13.20, Problem
1. Show that the sequence of measures /un * vn does not tend to 0 with respect to the
topology ^"i.

(b)    Let G be a locally compact group and let (^n), M be two sequences of real
bounded measures on G. Suppose that <.->//, in the topology ^~2, and that vn->v
in the topology ^"3 (notation of Section 13.20, Problem 1). Show that the sequence
fjLtt * vn tends to /x * v in the topology ^~2. (Observe that if /, g e ,^(G), the function
(x,y)\-*g(y?)f(xy) has compact support and can be uniformly approximated by a
linear combination of functions ut  vt, where u{ and vt are in JT(G).) Give an example
(with G = R) where /x = v = 0 and /Ltn * vn does not tend to 0 with respect to the
topology ^"3. Show that if /xn -> 0 with respect to ^"3 and if the sequence of norms
(W) is bounded, then /xn * vn -> 0 with respect to &"2. If /xn ~> yu with respect to ^"3,
and vn-v with respect to ^"3, show that \Ln * j/n ->^ * v with respect to ^"3 (similar
method).

(c)    With the same notation, show that if, in the topology ^6 ^n->ju, and vn-*v,
then \LK * vn -> ft * v (use Problem 2 of Section 13.20, and EgorofTs theorem).

(d)    Take G to be the group R2. Let a, b be the vectors of the canonical basis of G over
R; let pn be the measure on I = [0, TT] c: R whose density with respect to Lebesgue
measure is the function sin(2".x); let p,n be the measure pn  e0 on G, and let vn be the
measure fe/2 - 0 on G. Show that /xn->0 with respect to ^"6 and that yn~0 with
respect to ^3, but that the sequence (pn * vn) does not tend to 0 with respect to ^6.

5.    Let G be a compact group and /x a positive measure on G such that Suppfyz) = G,
and p * fj. = jit. Show that /x(G) = 1 and then that /x is a Haar measure on G. (Let

/e#V(G), and put g(x)  \f(yx) dfi(y), which is a continuous function on G. Show

that ff(x)=\ ff(yx) dp,(y), and deduce that g is constant, by considering the set of
points at which it attains its upper bound.)olvable if, for each